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Source. Erdős (1952), display (3), printed p. 379 (PDF, physical p. 1), with its qualification at the top of printed p. 380 (physical p. 2).

Domain qualification. The paper initially defines $P_x$ under a broad polynomial condition without excluding a zero running product or defining a greatest prime factor for zero. On printed p. 379 it immediately reduces to $f$ irreducible over $\mathbb Q$ of degree greater than one. This page records display (3) only in that explicitly labeled safe specialization, where the product has no zero factor and, for sufficiently large $x$, has absolute value greater than one. It does not attribute a new nonzero-product hypothesis to the source's broader wording.

Source-reported assertion. Within that safe specialization, after stating the theorem for which the paper gives a proof, Erdős says that a much more complicated method gives

$$P_x>x\exp\{(\log x)^{c_3}\}$$

for a positive constant $c_3=c_3(f)$. The next printed page opens with "(3) will not be proved in the present paper" (p. 380).

Credit boundary. This page records an historical assertion and the explicit omission of its proof from this paper. It is not a theorem record, an accepted result, a refutation of the paper's theorem, or the different formula $x^{c\log\log x}$. The same paragraph goes on to say that $P_x>c_4x^l$, with $l$ the degree of $f$, seems likely but, if true, must be very deep; that remark is likewise not proof evidence.

Relation to E976. The assertion concerns the running product in Problem 976, but receives no proved progress credit without a separate proof source.

Bears on. #976 (historical, explicitly unproved assertion only).

Living verification. Needs review. The opening domain convention, immediate irreducible reduction, exact displayed formula, and sentence withholding its proof were checked visually on physical pp. 1--2 / printed pp. 379--380 of the selected scan. No proof is supplied, reconstructed, or independently certified here.